The above set of equations can be embedded in a single non-linear Schrodinger like equation, using the so called Madelung transform [1]} defining the complex field \(\Phi (\vec{r},t)=\rho (\vec{r},t)^{1/2}\exp (iS(\vec{r},t)/\hbar )\) :
\(i\hbar \partial _t\Phi (\vec{r},t)=-\frac{\hbar ^2}{2m}\nabla ^2\Phi (\vec{r},t)+w(\vec{r},t)\Phi (\vec{r},t)\) 
